A unified way to build intervals that avoids empty and unphysical results near a boundary.The Feldman–Cousins construction is a method for building confidence intervals in particle physics that avoids two classic problems: empty intervals (when the observed data is so unlikely that the interval contains no values) and unphysical intervals (when the interval extends into forbidden regions, like negative masses). It provides a unified approach that works for both small and large signals, and it automatically handles the transition between upper limits and two-sided intervals. This makes it a standard tool for reporting results when the signal might be zero or very small.
Gary Feldman and Robert Cousins introduced this construction in 1998 in a paper titled 'A unified approach to the classical statistical analysis of small signals' (Physical Review D). It was a response to the then-common practice of using different methods for limits versus measurements, which led to inconsistencies.
The method uses a likelihood ratio to order possible outcomes. For each hypothesized true value of the parameter (e.g., the signal rate), you compute the likelihood of the observed data relative to the best-fit value. You then include in the confidence interval all parameter values for which the observed data falls within a certain probability mass (e.g., 90%) of the most likely outcomes, according to this ordering. This ordering is chosen so that the interval is never empty and never includes unphysical values. The result is a set of intervals that smoothly transition from upper limits (when the signal is weak) to two-sided intervals (when the signal is strong), without any ad hoc switching rules.
Widely used in particle physics for reporting limits on rare processes, such as neutrino masses, dark matter cross-sections, and searches for new particles, especially when the expected signal is small.
The Feldman–Cousins construction is sometimes called 'the unified approach' because it eliminates the need to decide beforehand whether you will report a limit or a measurement — the data itself determines the appropriate interval.
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